Representation Theoretical Methods in the Theory of Special Function

提议的研究涉及计算机探索和连接几个数学领域的信息传递，特别是表示理论，理论特殊函数和组合学。这些联系提供了一种强有力的发现工具，因为在一个领域非常明显的结果和机制往往会在另一个领域转化为非常重要和意想不到的事实。这类活动也非常适合培训年轻的研究人员，并为他们提供机会，让他们发现在计算机时代进行某些研究的方式。组合解释将数学信息，无论是代数的、分析的、逻辑的还是其他的，转化为视觉信息。在这样的环境下，即使是背景有限的学生也能体验到发现的乐趣。基于这些原因，PI计划继续通过研究生课程和研讨会将所有当前的研究成果直接带入课堂。事实上，在NSF的支持下，PI的学生和合作者的大部分工作都是由课堂活动产生的。代数组合学的子领域，也就是本奖项研究的重点，是由PI对麦克唐纳（1988）关于他现在著名的对称多项式基的猜想的表示理论方法创建的。20世纪90年代，PI和马克·海曼的联合工作导致了一个关于麦克唐纳多项式的空间“对角谐波”的Frobenius特征的推测公式。PI和海曼早期的计算机探索所产生的数据揭示了对角谐波与“停车函数”（计算机科学家创造的一种彩色组合结构）之间惊人的密切联系。这些发现导致了Haglund等人的Shuffle猜想的公式，该猜想给出了对角谐波的Frobenius特性，这是一个关于停车函数的漂亮的显式公式。2000年前后，马克·海曼用代数几何工具证明了与圆周率共同提出的原始猜想。这引起了代数几何学者对这一研究领域的注意。这种关注导致了这个领域真正令人惊讶的丰富，有各种各样的新工具和开放问题。最近，PI和他的合作者利用这些新发现，成功地建立了一个无限的“Shuffle猜想”族，将整个对称函数算子的李代数与新的“停车函数”类对象连接起来。在这一奖项的支持下，PI计划利用20年来在这一领域努力开发的大量工具，与他现在的博士生Emily Leven， Yeonkyung Kim和Marino Romero合作研究这些问题。PI和他现在的学生取得的初步结果非常有希望。事实上，最近Leven已经成功地解决了这些新猜想的无限亚族。历史上，难题一直是基础数学发现的源泉。在这方面，我们的具体调查领域不应例外。

The proposed research involves computer explorations and the transfer of information connecting several areas of mathematics, most particularly Representation Theory, the Theory Special Functions and Combinatorics. These connections provide a powerful vehicle of discovery, since results and mechanisms which may be quite obvious in one area often translate into highly nontrivial and unexpected facts in one of the other areas. This type of activity is also highly suitable for training young researchers and providing them with the opportunity to discover the manner in which some research can be carried in our Computer Age. Combinatorial interpretations translate mathematical information, be it algebraic, analytical, logical or otherwise, into visual information. Under this setting, even students with limited background can be brought to experience the joy of discovery. For these reasons the PI, plans to continue the practice of bringing all current research efforts right into the classroom through graduate courses and seminars. In fact most of the work of the students and collaborators of the PI carried out under prior NSF support resulted from such classroom activities. The subfield of Algebraic Combinatorics, that is the focus of the research carried out under this award was created by the PI's representation theoretical approach to the (1988) conjectures by Macdonald concerning his now-famous symmetric polynomial basis. The 1990's joint work of the PI and Mark Haiman led to a conjectured formula for the Frobenius characteristic of the space ``Diagonal Harmonics'' in terms of the Macdonald polynomials. Early computer explorations by the PI and Haiman yielded data which revealed a surprisingly intimate connection of Diagonal Harmonics with ``Parking Functions'' (a colorful combinatorial structure created by computer scientists.) These discoveries led to the formulation by Haglund et al. of the Shuffle Conjecture which gives the Frobenius Characteristic of Diagonal Harmonics a beautiful explicit formula in terms of Parking Functions. Around the year 2000, Mark Haiman proved the original conjectures formulated jointly with the PI by Algebraic Geometrical tools. This brought the attention of the Algebraic Geometers to this field of investigation. This attention resulted in a truly surprising enrichment of the field with a variety of new tools and open problems. Very recently the PI and his collaborators, using these new findings succeeded in formulating an infinite family of ``Shuffle Conjectures'' connecting a whole Lie Algebra of Symmetric Function Operators to new ``Parking Function'' like objects. Under the support of this award the PI plans to use the vast collection of tools developed in two decades of efforts in this area to work on these problems in collaboration with his present PhD students Emily Leven, Yeonkyung Kim and Marino Romero. Preliminary results obtained by the PI and his present students have been very promising. In fact, recently Leven has succeeded in solving an infinite subfamily of these new conjectures. Historically, difficult problems have been the source of fundamental mathematical discoveries. Our particular area of investigation should be no exception in this respect.